Mechanical advantage

Why does a small gear driving a large one trade away speed to gain force?

A car jack lets one person lift two tonnes. That looks like the machine is supplying strength the person does not have, and if it were, nothing would stop us from making a better jack until strength was free. Something evidently does stop us. So the interesting question is not how the jack multiplies force but what it must give up in exchange — and why the exchange is compulsory rather than a design compromise.

Start from what a machine could conceivably be doing. Work — in the mechanical sense of force applied along a distance — is the currency here, and energy is conserved: you cannot get more out than you put in. If a machine returned more force over the same distance, work would have appeared from nowhere. So if force goes up, distance must come down, in the same proportion. The trade is not chosen by an engineer. It is imposed before any engineer arrives.

Now watch it happen in gears, where the mechanism is unusually visible. Teeth mesh only if they are the same size on both wheels, so a wheel with twelve teeth driving one with thirty-six must turn three times for each turn of the larger. There is nothing subtle in that; you can count it. And because the teeth cannot slip past one another, the rims move together at the point of contact — so the same push at the mesh acts on a larger radius on the big wheel. Torque is force times radius, so the torque is multiplied by three, exactly the factor by which the speed fell.

Is that special to gears? Take a lever: both ends swing through the same angle about the fulcrum, so each end's travel is proportional to its arm length, and a four-times-longer effort arm moves four times as far under a quarter of the force. Take a block and tackle: if four rope segments share the load, each carries a quarter of it, and you must haul four metres of rope to raise the load one.

Three geometries, one identity. In the ideal case, output force divided by input force equals input distance divided by output distance, and both are fixed by geometry alone — tooth counts, arm lengths, rope segments. You can read a machine's mechanical advantage off a drawing without knowing anything about materials or loads.

So why say "ideal"? Because no real machine returns all the work you put in. Teeth slide against each other as they roll through the mesh, bearings drag, rope stiffens around a sheave, oil is churned. Every one of those takes work and gives back heat, so the work available at the output is less than the work spent at the input. Efficiency — useful work out over work in — is always below one, and the actual mechanical advantage of a machine you can hold is always smaller than the geometric number you calculated.

That loss is not uniform across designs, and where it is large it becomes a feature rather than a defect. A well-cut spur gear pair loses only a small percentage per stage. A worm drive, where a screw thread slides continuously along the wheel's teeth, can lose a great deal — and a sufficiently lossy worm drive cannot be driven backwards at all, because friction swallows the entire back-driving force. That is why hoists and some tuning pegs use them: a machine bad at returning work is a machine that holds its load when you let go.

Which leaves a corollary worth stating plainly, because it is the part that surprises people. A machine cannot reduce the work a job requires. Lifting the car takes the same work whether you use a jack, a ramp or your bare hands — rather more with the jack, in fact, once friction is counted. What a machine changes is the terms: it lets you supply that work in instalments small enough that a human arm can produce them, or in a direction a human arm can pull.

The most common misreading is that "mechanical advantage" means advantage in energy. It never does. It is a force ratio, and every gain in it is paid for in distance at exactly the same rate — so the only genuine advantage on offer is one of convenience and capability, not of total effort.

Two refinements. Power, being work per unit time, is what the ideal case conserves moment to moment: torque up and speed down leaves the product unchanged, which is why a gearbox does not change what an engine can deliver, only how it is presented to the wheels. And the geometric ratio assumes steady motion — accelerating a heavy gear train takes work of its own, so behaviour under changing speed is not captured by counting teeth.

• Why gear trains are built in stages, and what compounding the ratio costs in efficiency.
• How the same force-for-distance trade appears in electrical transformers and optical levers.

Every term the collection defines is gathered in the glossary.